It follows from [1]} that \(\Psi \in \mathcal {C}^1(\mathcal {U},\mathbb {R})\)  and it satisfies the Palais-Smale condition at every level \(c\in \mathbb {R}\) . Exactly the same argument used to prove [1]} shows that \(\mathrm {genus}(\mathcal {U})=\infty \)  (one simply replaces \(H_0^1(\Omega )^G\)  with \(H^1(\mathbb {R}^N)^{\phi _i}\) ). Now [1]} asserts that (REF ) has an unbounded sequence of solutions.
